Pulse mushrooms and garlic in a food processor until finely chopped. Heat oil in a 12-inch heavy skillet over medium-high heat until hot. Season chicken with salt and sauté, stirring occasionally, until just golden, about 3 minutes. Transfer with a slotted spoon to a bowl.
Reduce heat to medium and cook onion, stirring occasionally, until softened, about 3 minutes. Add mushroom mixture, rosemary, and cook stirring constantly until beginning to brown, about 4 minutes.
Add vinegar and cook until evaporated. Add chicken and tomatoes (with juice), then simmer, breaking up tomatoes with a spoon and stirring occasionally, until sauce is just thickened, about 25 minutes.
Meanwhile, cook orecchiette in a pasta pot of boiling water until al dente. Reserve 1 cup of boiling water before draining pasta.
Add reserved boiling water to mushroom sauce. Add arugula to sauce and stir until wilted. Stir in drained pasta and cook 1 minute. Season with rosemary salt and pepper to taste.